# Congress detail

**Authors**: Muro, Santiago.

**Resumen**: For each entire function $f$ on $n$ complex variables, there is a series of monomials, the monomial expansion of $f$, such that $$f(z)=sum_{alpha}a_alpha z^alpha,$$ for every $z$ and the convergence is uniform on each compact set. If $f$ is a holomorphic function on an infinite dimensional sequence space $X$, then it also has a monomial expansion, but in this case, the series does not necessarily converge for every $zin X$. There has been some effort to characterize the subset of $X$ where the monomial expansion of every holomorphic function on $mathcal F$ converge, where $mathcal F$ is a family of polynomials or of holomorphic functions on $c_0$ or $ell_p$. This set is called the {it set of monomial convergence} of $mathcal F$. The only case where set of monomial convergence has been completely characterized is when $X=ell_1$, or $mathcal F=mathcal P (^m c_0)$. In this talk we will describe the set of monomial convergence for the space $H_b(ell_r)$ of entire functions of bounded type on $ell_r$, for $1 < r leq 2$, and show that it is exactly a Marcinkiewicz sequence space. We will also talk about the set of monomial convergence for $mathcal P (^m ell_r)$, the space of $m$-homogeneous polynomials on $ell_r$, and for the space $H^infty(B_{ell_r})$ of bounded holomorphic on the unit ball of $ell_r$, $1 < r leq 2$. The talk is based on a joint work with Daniel Galicer (Universidad de Buenos Aires), Mart´in Mansilla (Universidad de Buenos Aires) and Pablo Sevilla-Peris (Universidad Polit´ecnica de Valencia).

**Meeting type**: Conferencia.

**Production**: Monomial expansions on sequence spaces.

**Scientific meeting**: Function Theory on Infinite Dimensional Spaces XVI.

**Meeting place**: Madrid.

**Organizing Institution**: Universidad Complutense de Madrid.

**It's published?**: No

**Meeting month**: 11